A STABILITY RESULT OF THE PÓSA LEMMA.

For an integer a and a graph G, the a-disintegration of G is the graph obtained from G by recursively deleting vertices of degree at most a until the resulting graph has no such vertex. Pósa proved that if a 2-connected graph contains a path on m ≥ k vertices with end-vertices in its ⌊(k-1)2]⌋disintegration, then G contains a cycle of length at least k. We prove that if a 2-connected graph contains a path on m ≥ k vertices with end-vertices in its ⌊(k-3)/2⌋-disintegration, then G contains either a cycle of length at least k or a specific family of graphs. As an application, we strengthen the Erdös--Gallai stablity theorem of Füredi, Kostochka, Luo, and Verstraëte. [ABSTRACT FROM AUTHOR]